Plan Groupoid cocycles and derivations · Plan Groupoid cocycles and derivations Jean Renault Universit e d’Orl eans September 7, 2012 1 The scalar case 2 Sauvageot’s construction

Plan

Groupoid cocycles and derivations

Jean Renault

Université d’Orléans

September 7, 2012

1 The scalar case

2 Sauvageot’s construction

3 The vector-valued case

J. Renault (Université d’Orléans) Cocycles and derivations — Yerevan 12 September 7, 2012 1 / 26

The scalar case

One-parameter automorphism groups

In the C*-algebraic formalism, time evolution of a physical system is givenby a pair (A, α), where A is a C*-algebra called algebra of observables andα = (αt)t∈R is a strongly continuous one-parameter group ofautomorphisms of A. The generator of α is a derivation, usuallyunbounded, which will be denoted by δ. The pair (A, α) will be calledhere a C*-system.

If A = B(H) is the algebra of bounded operators on a Hilbert space H,there exists an unbounded self-adjoint operator H on H such that

αt(A) = eitHAe−itH

Then the generator of α is the derivation δ where

δ(A) = i [H,A] = i(HA− AH).


The scalar case

Example 0

Suppose that A = Mn(C) and that α is given by a self-adjoint diagonaloperator H = diag(h1, . . . , hn). Then, for A = (Ak,l) ∈ Mn(C), we have:

αt(A)k,l = eit(hk−hl )Ak,l ; δ(A)k,l = i(hk − hl)Ak,l

Let us view the matrices as functions on G = {1, . . . , n} × {1, . . . , n} andrewrite these formulas after introducing γ = (k , l) ∈ G andc(γ) = hk − hl . This gives

αt(f )(γ) = eitc(γ)f (γ); δ(f )(γ) = ic(γ)f (γ)

Note that c : G → R satisfies the Chasles (or cocycle) relation:

c(k , l) + c(l ,m) = c(k,m).


The scalar case

Groupoids

The previous example fits into the general framework of groupoids andtheir convolution algebras.

A groupoid is a small category (G ,G (0)) where each arrow γ ∈ G isinvertible. The inverse is denoted γ−1. We denote by r(γ) the range of γand by s(γ) its source. A pair of arrows (γ, γ′) is composable iffs(γ) = r(γ′); then the composition is denoted by γγ′.

There are two examples to keep in mind: groups (this is the case whenG (0) is reduced to a singleton {e}) and equivalence relations (this is thecase when the map (r , s) : G → G (0) × G (0) is injective). In that case,arrows are pairs (x , y); composition law and inverse are respectively

(x , y)(y , z) = (x , z); (x , y)−1 = (y , x)


The scalar case

Convolution algebra

We assume that G has locally compact Hausdorff topology compatiblewith its algebraic structure and a Haar system (λx)x∈G (0) where for allx ∈ G (0), λx is a Radon measure on the fibre G x = r−1(x) and we have

(invariance) γλs(γ) = λr(γ) ;

(continuity) x →∫fdλx is continuous for all f ∈ Cc(G ) .

We define the C*-algebra C ∗(G ) as the C*-completion of the ∗-algebraCc(G ) of continuous compactly supported functions on G , where

f ∗ g(γ) =∫

f (γγ′)g(γ′−1)dλs(γ)(γ′)

f ∗(γ) = f (γ−1).


The scalar case

Scalar cocycles and derivations

A map c : G → R is a cocycle if

c(γγ′) = c(γ) + c(γ′)

When G = X × X (or any equivalence relation), this is above Chaslesrelation.

We assume that c is continuous and we define δc : Cc(G )→ Cc(G ) by

δc(f )(γ) = ic(γ)f (γ).

It is immediate that δc is a derivation:

δc(f ∗ g) = δc(f ) ∗ g + f ∗ δc(g).


The scalar case

Some properties

Defineαt(f )(γ) = e

itc(γ)f (γ).

It is a one-parameter automorphism group of C ∗(G ). This shows that δcis a pregenerator, in particular it is closable.

Proposition

Let G , c , δc be as above. Then,

1 δc is bounded iff c is bounded;

2 δc is inner if c is a coboundary.

Recall that a cocycle c : G → R is a coboundary if there existsb : G (0) → R such that c = b ◦ r − b ◦ s. Here, we are dealing withcontinuous cocycles and we insist on having b continuous.


The scalar case

Example 1

This example is a groupoid description of the Ising model. As usual, Λ is alattice in Rd and each site carries a spin ±1. The configuration space isX = {−1,+1}Λ. Let G be the graph of the equivalence relation on Xwhere two configurations are equivalent iff the number of sites where theydiffer is finite. The C*-algebra A of observables is an algebra of matricesindexed by G : its elements are functions on G and the operations are theusual matrix multiplication and involution:

ab(x , z) =∑y

a(x , y)b(y , z); a∗(x , y) = a(y , x)

The one-parameter automorphism group α is given by

αt(a)(x , y) = eitc(x ,y)a(x , y)

where the energy cocycle c : G → R is defined by

c(x , y) = −∑i ,j

J(i , j)[x(i)x(j)− y(i)y(j)]

where J depends on the nature of the interaction.J. Renault (Université d’Orléans) Cocycles and derivations — Yerevan 12 September 7, 2012 8 / 26

The scalar case

Example 2

The Cuntz algebra Od is the C*-algebra generated by d isometriesS1, . . . ,Sd on a Hilbert space H whose ranges are mutually orthogonal andspan H.

The gauge automorphism group is the one-parameter automorphism groupα such that αt(Sk) = e

itSk .

Its groupoid description is Od = C ∗(G ), where

G = {(x ,m − n, y) : x , y ∈ X ,m, n ∈ N,T nx = Tmy}

where T is the one-sided shift on X = {1, . . . , d}N.

The above gauge automorphism group α is then given by the gaugecocycle c : G → R such that c(x , k , y) = k.


The scalar case

Example 3

One usually defines a tiling Tas a subdivision of the plane R2 into tiles.Here is a famous example of a tiling (or rather a patch of this tiling):


The scalar case

Example 3 (cont’d)

We denote by T + x the translate of the tiling T by the vector x ∈ R2.We define a metric on the set of tilings: T and T ′ are � close if there existvectors x and x ′ of norm ≤ � such that T + x and T ′ + x ′ agree on theball of centre 0 and radius 1/�.

The hull of a tiling T0 is the orbit closure

Ω(T0) = T0 + R2

The group R2 acts continuously on Ω(T0) and one can form semi-directgroupoid

G (T0) = {(T , x ,T ′) ∈ Ω(T0)× R2 × Ω(T0) : T + x = T ′}


The scalar case

Example 4 (cont’d)

There is a natural cocycle on G (T0), namely c : G (T0)→ R2 such thatc(T , x ,T ′) = x . This construction provides a C*-system (C ∗(G (T0)), α).

What is this C*-system good for?

Aperiodic tilings model quasicrystals. The corresponding C*-algebra givesinformation on the spectrum of its Schrödinger Hamiltonian, in particularon its gaps. A large part of the work is to compute the K-theory ofC ∗(G (T0)) and its image under a canonical trace.


Sauvageot’s construction

Non-commutative Dirichlet forms

J.-L. Sauvageot has given in 1989 the following construction in theframework of non-commutative Dirichlet forms:

Let (Tt)t≥0 be a strongly continuous semi-group of completely positivecontractions (also called Markov operators) of a C*-algebra A. We denoteby −∆ its generator (so that Tt = e−t∆); it is a complete dissipation. Weassume that we have a dense sub ∗-algebra A which is an essential domainof the generator.

The associated Dirichlet form is the sesquilinear map L : A×A → Adefined by

L(a, b) = 12

[a∗∆(b) + ∆(a∗)b −∆(a∗b)].

Then L is completely positive in the sense that

∀ n ∈ N∗, ∀ a1, . . . , an ∈ A, [L(ai , aj)] ∈ Mn(A)+.



C*-correspondence

Before giving Sauvageot’s representation theorem, let us recall thedefinition of a correspondence in the C*-algebraic framework.

Definition

Let A and B be C∗-algebras. An (A,B)-C*-correspondence is a rightB-C∗-module E together with a ∗-homomorphism π : A→ LB(E).

The inner product E × E → B will be denoted < ·, · >.



GNS type construction

The positivity of the non-commutative Dirichlet form gives the followingGNS representation:

Theorem (J.-L. Sauvageot, 1989)

Let (Tt)t≥0 be a semi-group of CP contractions of a C*-algebra A asabove and let L be its associated Dirichlet form. Then

1 There exists an (A,A)-C*-correspondence E and a derivationδ : A → E such that

for all a, b ∈ A, L(a, b) =< δ(a), δ(b) >;the range of δ generates E as a C*-module.

2 If (E ′, δ′) is another pair satisfying the same properties, there exists aC*-correspondence isomorphism u : E → E ′ such that δ′ = u ◦ δ.



Classical example: the heat equation semi-group

Here the C*-algebra is A = C0(M), where (M, g) is a completeRiemannian manifold, Tt = exp(−t∆) is the heat equation semi-groupand ∆ is the Laplacian. The Dirichlet form is

L(f , g) = 12

[f ∆(g) + ∆(f )g −∆(f g)].

Let us introduce the complex tangent bundle TCM, the correspondingHilbert C*-module C0(M,TCM) over C0(M), where the inner product isgiven by < ξ, η > (x) = gx(ξ(x), η(x)), and the gradient

∇ : C∞c (M)→ C0(M,TCM).

It is a derivation and we have

L(f , g) =< ∇f ,∇g > .

Thus this is the derivation given by the Sauvageot construction. Oneretrieves the tangent bundle and the gradient from the laplacian only. Thisis why Sauvageot calls E the tangent bimodule in the general case.


Vector-valued cocycles

Groupoid example: conditionally negative type functions

I want to present another example of Sauvageot’s construction. Instead ofthe commutative C*-algebra C0(M), we shall consider a groupoidC*-algebra C ∗(G ). Instead of the Laplacian ∆, we shall consider theoperator of pointwise multiplication by a conditionally negative typefunction ψ on G .

Definition

Let G be a groupoid. A function ψ : G → R is said CNT (conditionally ofnegative type) if

1 ∀x ∈ G (0), ψ(x) = 0,2 ∀γ ∈ G , ψ(γ−1) = ψ(γ),3 ∀n ∈ N∗, ∀ζ1, . . . , ζn ∈ R such that

∑n1 ζi = 0,

∀x ∈ G (0), ∀γ1, . . . , γn ∈ G x ,∑i ,j

ψ(γ−1i γj)ζiζj ≤ 0.



CNT functions define dissipations

Proposition

Let (G , λ) be a locally compact groupoid with Haar system and letψ : G → R be a continuous CNT function. Then,

1 Pointwise multiplication by e−tψ, where t ≥ 0 defines a completelypositive contraction Tt : C

∗(G )→ C ∗(G ) and (Tt)t≥0 is a stronglycontinuous semi-group of completely positive contractions of C ∗(G ).

2 Its generator −∆ has the dense sub-∗algebra A = Cc(G ) as essentialdomain and ∆f = ψf .

Exercise. Compute the Sauvageot’s derivation associated with thissemi-group.



CNT functions admit cocycle representations

Proposition (J.-L. Tu, 1999; R 2012)

Let (G , λ) be a locally compact groupoid with Haar system. Letψ : G → R be a continuous function conditionally of negative type. Then

1 There exists a pair (E , c) consisting of a continuous G -Hilbert bundleE and a continuous cocycle c : G → r∗E such that

for all γ ∈ G , ψ(γ) = ‖c(γ)‖2;for all x ∈ G (0), {c(γ), γ ∈ G x} is total in Ex .

2 If (E ′, c ′) is another pair satisfying the same properties, there exists aG -equivariant isometric continuous bundle map u : E → E ′ such thatc ′ = u ◦ c.



What is a G -Hilbert bundle?

Definition

A bundle consists of topological spaces E and X and a continuous,open and surjective map π : E → X .A bundle π : E → X is a Hilbert bundle if each fiber Ex = π−1(x) is aHilbert space and

1 addition, scalar multiplication and norm are continuous;2 if (ui ) is a net in E such that ‖ui‖ → 0 and π(ui )→ x , then ui → 0x .

Let G be a topological groupoid with G (0) = X . A G -Hilbert bundleis a Hilbert bundle π : E → X endowed with a continuous, linear andisometric action of G .

Linear and isometric means that for all γ ∈ G , L(γ) : Es(γ) → Er(γ) is alinear isometry.



What is a vector-valued cocycle?

Definition

Let E → G (0) be a G -Hilbert bundle. A cocycle for E is a continuoussection c : G → r∗E (this means that c(γ) ∈ Er(γ)) such that

c(γγ′) = c(γ) + L(γ)c(γ′).



The crossed-product C*-correspondence

Given a G -Hilbert bundle E , we let Cc(G , r∗E ) be the space of compactly

supported continuous sections of r∗E .We define the left and right actions: for f , g ∈ Cc(G ) and ξ ∈ Cc(G , r∗E ),

f ξ(γ) =

∫f (γ′)L(γ′)ξ(γ′

−1γ)dλr(γ)(γ′)

ξg(γ) =

∫ξ(γγ′)g(γ′

−1)dλs(γ)(γ′)

and the right inner product: for ξ, η ∈ Cc(G ),

< ξ, η > (γ) =

∫(ξ(γ′

−1)|η(γ′−1γ))s(γ′)dλr(γ)(γ′).

This can be completed into a C*-correspondence C ∗(G , r∗E )



Construction of the derivation

Let now c : G → r∗E be a continuous cocycle. Defineδc : Cc(G )→ Cc(G , r∗E ) by

δc(f )(γ) = ic(γ)f (γ).

Proposition

1 δc is a derivation: δc(f ∗ g) = δc(f )g + f δc(g);2 δc : Cc(G )→ C ∗r (G , r∗E ) is closable.



Solution of the exercise

Theorem (R 2012)

Let (G , λ) be a locally compact groupoid with Haar system and letψ : G → R be a continuous CNT function. Then the derivation (E , δ)associated with the semi-group (Tt) of pointwise multiplication by e

−tψ bySauvageot is (C ∗(G , r∗E ), δc), where (E , c) is the cocycle representing ψ.

For the proof, one checks that the Dirichlet form of (Tt)t≥0 has thedesired expression:

L(f , g) =< δc(f ), δc(g) >

and that the range of δc generates C∗(G , r∗E ) as a C*-module.



Remarks

Just as in the scalar case, δc is bounded iff c is bounded. It is inner ifc is a continuous boundary.

The theory of unbounded derivations with values in (M,M)-Hilbertmodules, where M is a von Neumann algebra (usually equipped witha finite trace) is well developed. This is not the case for unboundedderivations with values in (A,A) C*-bimodules. The above groupoidexample may be useful to develop this theory.

It would be nice to illustrate the vector-valued case by physicalexamples.



The End

Thank you for your attention!


The scalar caseSauvageot's constructionVector-valued cocycles

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Plan Groupoid cocycles and derivations · Plan Groupoid cocycles and derivations Jean Renault Universit e d’Orl eans September 7, 2012 1 The scalar case 2 Sauvageot’s construction